oooo pixel art itself really is like a lost art form. I've messed around it but haven't fully dedicated myself to it since i'm still focusing on nailing down digital art. All the tutorials i've been able to find about it are like from obscure websights and forums posts from like 10ish years ago. I could link them to you if you'd like if that's also something you're interested in.

I don't know if there's really any sort of program that's dedicated to the animation of pixel art itself but i know there are a few programs with that feature kind of somewhat built in.

I think Pro Motion is a program that some professionals use? From what i recall it has a pretty decent animation program built into it.

Graphics Gale is another program i used that seems to have a very user friendly UI and does in fact have an animation program built into it as well but from what i recall its not to the extent that Pro Motion has.

You could also use programs like MSPaint, Photoshop and even Paint Tool Sai to create pixel art and export the .png files into a separate animation program like flash to have them animated probably.

Its been a few years since I've messed with Pro Motion/Graphics Gale so i could be off but i think those might be good places to start.

I've done some pixel animations, I used Photoshop mostly, set to nearest neighbor, set pencil to 1 pixel, zoom right in and start pounding pixels. Photoshops animation timeline is great for animated gifs. I've also used After Effects for more heavy animations. do the frames in Photoshop and import and compose in After Effects.

Though it seems a lot of pixel artists don't recommend Photoshop for pixel art, I'm not exactly sure why.

There is a lot of free programs out there, but I really haven't used any of them. When I first started I found http://2dwillneverdie.com/ to be a very useful resource.

I could give you my two cents on the subject. Although I'll warn you that I'm rather opinionated. What works for me might not work for you. (And vice versa.)

First, you should never save your finished pixel art as a jpeg. Use a gif or png. And of course, you should never upload a bmp to the internet.

I personally do my pixel art in ms paint. For most pixel art you don't really need all the bells and whistles from high end art programs. Tasks are often just faster to do in paint. It also leaves a tiny footprint. (I have windows 7 but I use the xp version of paint. People that use paint hate the new version of it found in win 7, 8, etc.)

I've heard good things about Graphics Gale. But I've never used it myself.

I rarely make tiles or character animation frames bigger then 50x50 pixels. I also prefer to create all my animation frames in paint first, then import them into a basic gif animation program or into MMF2. (The program I use to make my game.)

This is a very different process then trying to animate frames within an art program it's self. So your preference for the animation method, how large the sprite is, how many frames, etc. might effect which art program is best for you.

Older versions of paint don't have layers, which is fine for me because I wouldn't use them anyway.

My bmps kinda look like a huge pile of Legos poured out on the floor; body parts and severed limbs all over the place. Then I copy and paste these parts together, pixel perfect. Using layers to achieve the same effect would actually take longer. (Although some people still prefer to use them.)

If I need something that I haven't already drawn in the past, I create it then and there.

My pixel art folder is just over 3 GB. When doing pixel art you should always create back ups of your frames during the entire animation. Using my profile gif as an example; I have frames of her with no head, no arms, no torso, etc. Additionally, I have all of her frames done with alternate skin tones.

Pixel art takes a lot of time. It's not strange to spend 3 hours on a non animated 16x16 tile or item icon. You tweak it, tweak it some more, then tweak it again. Just like traditional art, if you don't give your all you won't improve. I'm not the best, but I'm better then I was last year. I think that's the proper mindset to have.

Never edit (copy and modify someone else's sprites) or rip (steal and use someone else's sprites as is). It's the same as tracing illustrations. Don't do it.

Never use the blur tool on pixel art.

If you are going to be animating sprites, study up on simple cartoon animations.